Free Electron Model for Metals

  Free Electron Model for Metals • Metals are very good at conducting both heat and electricity.

• Metals were described as behaving like a set of nuclei forming a lattice with a “sea of electrons” shared between all nuclei/lattice (moving freely between them): This is referred to as the free electron model for metals.
• This model explains many of the properties of metals:

  – Electrical Conductivity: The mobile electrons carry current.

• – Thermal Conductivity: The mobile electrons can also carry heat.
• – Malleability and Ductility: Deforming the metal still leaves each cation surrounded by a “sea of electrons”, so little energy is required to either stretch or flatten the metal.
• – Opacity and Reflectance (Shininess): The electrons will have a wide range of energies, so can absorb and re-emit many different wavelengths of light.

The Free Electron Gas Model

  Plot U(x) for a 1-D crystal lattice : Simple and crude finite-

  U square-well model:

  U = 0

Can we justify this model? How can one replace the entire lattice by a constant (zero) potential? Free-Electron Model

   

  m k m p

  2

  2 ^{2 2 2}   

   classical description

   

         E m

  

2

^{2 2}    z k y k x k L xyz _{z y x} sin sin sin

  8 ₃    

  In a 3D slab of metal, e’s are free to move but must remain on the inside Solutions are of the form:

  L n _z 

  2

  2

  2

  

2

h

  With energy:

  Quantum Mechanical Viewpoint

  At T = 0, all states are filled up to the Fermi energy

  2 h ₂

₂

₂

   n  n  n  _{Fo  x y z  2 max} mL

  2

  2

  2

  2 max n  n  n  n x y z total number of states up to an energy e

  fermi

  . 10 646

  _Fo h  

  8 n mL

    ^{max 2}

²

²

  V N m

  V N m eV x

  

       

     

  8    

  3

  

3

  N 3 / ² _{2 19} ^{3 / 2} ₂ .

  :

   









    

  2 n sphere of volume

  2

  1

  8

  1

  8

  4

  3 max

  3

Fo h

  Example: Numerical Values for Copper slab N _{3 3 3}

  = 8.96 gm/cm 1/63.6 amu 6e23 = 8.5e22 #/cm = 8.5e28 #/m

  V _{2 / 3 2 / 3}

  2   h N

  N

  3  

   ^{19 2} x eV m eV

     _Fo 3 . 646 10 .  7 .

      m

  V V 8 

     

  n = 4.3 e 7

  max

  so we can easily pretend that there’s a smooth distrib of n n n -states

  x y z Density of States How many combinations of are there within an energy interval e to e + de ? _{3 / 2 2}

  3

  2

  

   ^

          d g f KE Tot

  V dN 

  3

   2 / ^{1 2 / 1 2 / 1 3}

  

    

    

    

  3   

  3

  8

  8   

  8

  V dN

₂

^{2 / 1} ₂

  V N  dE h

m

h mE

  h mE

  

    

    

    

  3   

  8

   2 / ³ ₂

  V N m _Fo h 

     

  Huge number of states, it can be treated in continuous energy

  Electron follow Fermi-Dirac Distribution

   

  1

  1 ) ( _/   _{ kT f E E} e E f At T ≠ 0 the electrons will be spread out among the allowed states

  How many electrons are contained in a particular energy range?    

   









    occuring energy this of y probabilit energy particular a have to ways of number

   

  1

  1

  2

  8 ) ( _{/ ) (} ^{2 / 1 2 / 1 3} ₃  

   kT _f ^E e E m h

  V E n 

   ) ( ) ( ) (

  E f E g E n 

• Electrical transport (relaxation time) in conductor

     

     

  

F

 

  V N n  ty conductivi Fermi velocity v density electron n path free mean electron l time collision time relaxation

  

     

  

  E j Law Ohm

• ^{2 *}
²

  F ^{F r} v l m e n v l v l m e n

  E Electron Conductivity

  2 

    

      j nev ne m d

  eE v m dt dv m ^{d d}

   

Simple Kinetic Theory of Heat Conduction

  1

  1 _{Hot q x Cold} q  nEv  nEv     x x x  v x x  v _{x x} τ τ

  2

  2 _{v t x x x} d nEv

  ( )

_x

• q  v  _{x x} dx
_, Taylor Expansion: d nE ₂ ( ) _{2 du dT}   - v

   _x ^v _x  _{dT dx}

• dx

  local thermodynamics equilibrium: u=u(T)

  C du dT ₂  / _{2 2} v dT q C

   

  

• v v _x

  / 3 x dx

  3 ₂ 

   Cv  Cv  thermal conductivi ty k 

  3

  3

  NE U

  

Assigments

• Proof that at T=0 the total energy of electron
• Calculate specific heat capacity using FEM
• What fraction of free electron in Cu have a kinetic energy between 3.95 eV and 4.05 eV at room temperature

  ^Fo

  5

  3 

Heat Capacity of the Quantum-Mechanical FEM

  Quantum mechanics showed that the occupation of electron states is governed by the Pauli exclusion principle, and that the probability of occupation of a state with energy E at temperature T is:

  1 where  = chemical potential  E for kT << E f E 

  ( ) _{E  / kT} ^{F F}    e 

  1 _{1 IS TR D IB U TI O} N _{0.6 0.8 1000 K F 300 K R E I- M D IR A} C _{0.2 0.4 100 K}

• -0.1 -0.05
_{E-  (eV)} ^{0.05 0.1}

Heat Capacity of Metals: Theory vs. Expt. at low T

  Very low temperature measurements reveal: Meta / =   _{expt FEG}

  Results for simple _{l m*/m}   _{expt FEG} metals (in units _{Li 1.63 0.749 2.18} The discrepancy is mJ/mol K) show _{Na 1.38 1.094 1.26} “accounted for” by that the FEG values _{K 2.08 1.668 1.25} defining an effective are in reasonable electron mass m* that is agreement with _{Cu 0.695 0.505 1.38} due to the neglected experiment, but are _{Ag 0.646 0.645 1.00} electron-ion interactions always too high: _{Au 0.729 0.642 1.14}

   

  Problems with Free Electron Model

Other Problems with the Free Electron Model

• graphite is conductor, diamond is insulator
• variation in colors of x-A elements
• temperature dependance of resistivity
• resistivity can depend on orientation of crystal & current I direction
• frequency dependance of conductivity
• variations in Hall effect parameters
• resistance of wires effected by applied B-fields • .
• .
• .

  

Nearly-Free Electron Model

version 1 – SP221

k   / ² _{a   / 2  / 2 k}  ₂ 

  2

  This treatment assumes that when a reflection occurs, it is 100%.

  Nearly-Free Electron Model version 2 – SP324

• Bloch Theorem • Special Phase Conditions, k = +/- m p/a
• the Special Phase Condition k = +/- p/a

  (x) ~ u e

  i(kx-wt)

  (x) ~ u(x) e

  i(kx-wt)

   

  ~~~~~~~~~~ amplitude

  In reality, lower energy waves are sensitive to the lattice: Amplitude varies with location

  Bloch’s Theorem u(x+a) = u(x) (x+a) e

• i(kx+ka-wt)
• i(kx-wt)

   (x) e

  (x) ~ u(x) e

  i(kx-wt)

  (x+a)  e

  ika

  (x)

  Something special happens with the phase when

  e

  ika

  = 1 ka = +/ m p m = 0 not a surprise m = 1, 2, 3, …

  ... , 2 ,

  a a k  

    

  a k    Consider a set of waves with +/ k-pairs, e.g.

  k = + p/a moves  k =  p/a moves  This defines a pair of waves moving right & left

  Two trivial ways to superpose these waves are: 

  ikx

  ikx

  

  

  ~ e

  ikx

   e

  ikx

• ~ e
• e
• ~ 2 cos kx

  

  

  

  ~ 2i sin kx

  

• ~ 2 cos kx

  

  

  ~ 2i sin kx |

  2

• |

  ~ 4 cos

  2

  kx |

  

  |

  2

  ~ 4 sin

  2

  kx Free-electron Nearly Free-electron

  Kittel

  2

  

Effective Mass m*

A method to force the free electron

model to work in the situations where

there are complications

  

2

m k

     free electron KE functional form

• 2
Effective Mass m*
• describing the balance between applied ext-E and lattice site reflections

  2

  2

  2

  

1

   k m  

  



• 1

  m* a =

  S

  F

  ext

  q E

  ext

  2)

  greater curvature, 1/m* > 1/m > 0,  m* < m  net effect of ext-E and lattice interaction provides additional acceleration of electrons m = m* greater |curvature| but negative,

  At inflection pt net effect of ext-E and lattice interaction de-accelerates electrons

  No distinction between m & m*,

  2

  2

  2

• 2

  2

  2 m k m k ^{lattice from on perturbati apply}

      

  Another way to look at the discontinuities

  Shift up implies effective mass has decreased, m* < m, _{allowing electrons to increase their speed and join faster electrons in the band. The enhanced e-lattice interaction speeds up the electron.}

  Shift down implies effective mass has increased, m* > m, _{prohibiting electrons from increasing their speed and making them become similar to other electrons in the band. The enhanced e-lattice interaction slows down the electron} From earlier:

  Even when above barrier, reflection and transmission coefficients can

increase and decrease depending upon the energy. change in motion due to reflections is more significant than change in motion due to applied field

change in motion

due to applied field

enhanced by change in reflection coefficients

Nearly-Free Electron Model version 3 à la Ashcroft & Mermin, Solid State Physics

  This treatment recognizes that the reflections of electron waves off lattice sites can be more complicated. A reminder:

Waves from the left behave like: _{iKx iKx}

  left ^{the from} e r e

      ^iKx _{left the} ^from e t

    K ^{2 2}

   Waves from the right behave like: _{iKx iKx}

  right ^{the from} e r e

      ^iKx _{right the from} e t

    

  2 ²

  right left sum B A     

  Bloch’s Theorem defines periodicity of the wavefunctions:     x e a x sum ika sum

          x e a x sum ika sum

      unknown weights

          x e a x _sum ^ika _sum

          x e a x _sum ^ika _sum

  Applying the matching conditions at x   a/2

  A + B left ^right

  A + B left ^right

  A + B left ^right

  A + B left ^right iKa iKa e t e t r t ka

  





  

  2

  1

  2 cos

  2

  ^{2 2} 

    And eliminating the unknown constants A & B leaves: For convenience (or tradition) set:

  2

  2 1 r t  

   i e t t

    _i

   e r i r    ka t

  Ka cos cos

    

  Ka   cos

     ka cos t

  Related to Related to Energy _{2 2} possible Lattice spacings

   K   m

  2

  

ons

gi

re

ion

ut

sol

ed

low

al